The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed \(\varepsilon \in (0,1)\), computes a \((1-\varepsilon )\)-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.

最大旅行商问题 (Max TSP) 包括在给定的完整加权图中找到具有最大边总权重的哈密顿循环。这个问题在一般度量情况下是 APX 难的，但在几何设置中允许多项式时间近似方案，当边缘权重由固定维实空间中的向量范数引起时。我们在固定加倍维度的任意度量空间中提出了最大 TSP 的第一个近似方案。所提出的算法实现了一个高效的 PTAS，对于任何固定的\(\varepsilon \in (0,1)\)，计算\((1-\varepsilon )\)- 在三次时间内问题的近似解。此外，我们建议使用三次时间算法，该算法在固定和次对数加倍维度中找到度量 Max TSP 的渐近最优解。